The gradient of a smooth curve, , at a point is the gradient of the tangent to the curve at the point . Point is on the curve and is a neighbouring point whose value is displaced a small quantity, .

The idea behind differentiation is that as becomes very small, the gradient of tends towards the gradient of the curve. In the limit as becomes infinitesimally close to zero, the gradient becomes the gradient of the curve.

We write:

there is a fair bit of analytic work missing (higher education) to make these ideas sound.

We also write:

STANDARD RESULTS

Standard results can be proved for different functions.

If then

If , then we need to consider the small angle approximation that is if radians is very small (infinitesimal), then and , and compound trigonometry from which follows,

The differentiation process described above is linear and extends to more complicated functions. That is to say that if, where ,

There’s always space on the inter-web for another proof of Pythagoras’s Theorem. Here’s one that uses the following equal areas property of parallelograms.

This kind of area chopping and shape translation is a feature of Euclidean geometry and our senses support it’s veracity at the order of size of the classroom.

The squares on the sides of a right-angle triangle set up a system of parallel lines which can then be used to demonstrate the Theorem using the above equal areas property.

The thread does not stop here though. Taking the parallel line structure which makes this work we get a generalisation of Pythagoras to non-right-angled triangles with the area of the parallelogram on the longest side being the sum of the areas of those constructed on the other two sides.